Stable random variables with a complex index $\alpha$. The case of $|\alpha - 1/2|<1/2$

The case of $|\alpha - 1/2|<1/2$. In this paper, we construct complex-valued random variables that satisfy the usual stability condition, but for a complex parameter $\alpha$ such that $|\alpha-1/2|<1/2$. The characteristic function of the obtained random variables is found and limit theorems for sums of independent identically distributed random variables are proved. The corresponding Lévy processes and semigroups of operators corresponding to these processes are constructed.